3.3.46 \(\int \frac {\arctan (a x)}{x (c+a^2 c x^2)^{5/2}} \, dx\) [246]

3.3.46.1 Optimal result

Integrand size = 22, antiderivative size = 279 \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {a x}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 a x}{9 c^2 \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\arctan (a x)}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{c^2 \sqrt {c+a^2 c x^2}} \]

output

-1/9*a*x/c/(a^2*c*x^2+c)^(3/2)+1/3*arctan(a*x)/c/(a^2*c*x^2+c)^(3/2)-11/9* 
a*x/c^2/(a^2*c*x^2+c)^(1/2)+arctan(a*x)/c^2/(a^2*c*x^2+c)^(1/2)-2*arctan(a 
*x)*arctanh((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c* 
x^2+c)^(1/2)+I*polylog(2,-(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/ 
2)/c^2/(a^2*c*x^2+c)^(1/2)-I*polylog(2,(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a 
^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)
 

3.3.46.2 Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.60 \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\left (1+a^2 x^2\right )^{3/2} \left (-\frac {45 a x}{\sqrt {1+a^2 x^2}}+\frac {45 \arctan (a x)}{\sqrt {1+a^2 x^2}}+3 \arctan (a x) \cos (3 \arctan (a x))+36 \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )-36 \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )+36 i \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-36 i \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\sin (3 \arctan (a x))\right )}{36 c \left (c \left (1+a^2 x^2\right )\right )^{3/2}} \]

input

Integrate[ArcTan[a*x]/(x*(c + a^2*c*x^2)^(5/2)),x]
 

output

((1 + a^2*x^2)^(3/2)*((-45*a*x)/Sqrt[1 + a^2*x^2] + (45*ArcTan[a*x])/Sqrt[ 
1 + a^2*x^2] + 3*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 36*ArcTan[a*x]*Log[1 - E 
^(I*ArcTan[a*x])] - 36*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] + (36*I)*Pol 
yLog[2, -E^(I*ArcTan[a*x])] - (36*I)*PolyLog[2, E^(I*ArcTan[a*x])] - Sin[3 
*ArcTan[a*x]]))/(36*c*(c*(1 + a^2*x^2))^(3/2))
 

3.3.46.3 Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {5501, 5465, 209, 208, 5501, 5465, 208, 5493, 5489}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\arctan (a x)}{x \left (a^2 c x^2+c\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 5501

\(\displaystyle \frac {\int \frac {\arctan (a x)}{x \left (a^2 c x^2+c\right )^{3/2}}dx}{c}-a^2 \int \frac {x \arctan (a x)}{\left (a^2 c x^2+c\right )^{5/2}}dx\)

\(\Big \downarrow \) 5465

\(\displaystyle \frac {\int \frac {\arctan (a x)}{x \left (a^2 c x^2+c\right )^{3/2}}dx}{c}-a^2 \left (\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{5/2}}dx}{3 a}-\frac {\arctan (a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 209

\(\displaystyle \frac {\int \frac {\arctan (a x)}{x \left (a^2 c x^2+c\right )^{3/2}}dx}{c}-a^2 \left (\frac {\frac {2 \int \frac {1}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 c}+\frac {x}{3 c \left (a^2 c x^2+c\right )^{3/2}}}{3 a}-\frac {\arctan (a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 208

\(\displaystyle \frac {\int \frac {\arctan (a x)}{x \left (a^2 c x^2+c\right )^{3/2}}dx}{c}-a^2 \left (\frac {\frac {2 x}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {x}{3 c \left (a^2 c x^2+c\right )^{3/2}}}{3 a}-\frac {\arctan (a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 5501

\(\displaystyle \frac {\frac {\int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \int \frac {x \arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx}{c}-a^2 \left (\frac {\frac {2 x}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {x}{3 c \left (a^2 c x^2+c\right )^{3/2}}}{3 a}-\frac {\arctan (a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 5465

\(\displaystyle \frac {\frac {\int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \left (\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {\frac {2 x}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {x}{3 c \left (a^2 c x^2+c\right )^{3/2}}}{3 a}-\frac {\arctan (a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 208

\(\displaystyle \frac {\frac {\int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \left (\frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {\frac {2 x}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {x}{3 c \left (a^2 c x^2+c\right )^{3/2}}}{3 a}-\frac {\arctan (a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 5493

\(\displaystyle \frac {\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{x \sqrt {a^2 x^2+1}}dx}{c \sqrt {a^2 c x^2+c}}-a^2 \left (\frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {\frac {2 x}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {x}{3 c \left (a^2 c x^2+c\right )^{3/2}}}{3 a}-\frac {\arctan (a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )\)

\(\Big \downarrow \) 5489

\(\displaystyle -a^2 \left (\frac {\frac {2 x}{3 c^2 \sqrt {a^2 c x^2+c}}+\frac {x}{3 c \left (a^2 c x^2+c\right )^{3/2}}}{3 a}-\frac {\arctan (a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )+\frac {-a^2 \left (\frac {x}{a c \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)}{a^2 c \sqrt {a^2 c x^2+c}}\right )+\frac {\sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{c \sqrt {a^2 c x^2+c}}}{c}\)

input

Int[ArcTan[a*x]/(x*(c + a^2*c*x^2)^(5/2)),x]
 

output

-(a^2*((x/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*x)/(3*c^2*Sqrt[c + a^2*c*x^2])) 
/(3*a) - ArcTan[a*x]/(3*a^2*c*(c + a^2*c*x^2)^(3/2)))) + (-(a^2*(x/(a*c*Sq 
rt[c + a^2*c*x^2]) - ArcTan[a*x]/(a^2*c*Sqrt[c + a^2*c*x^2]))) + (Sqrt[1 + 
 a^2*x^2]*(-2*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]] + I*Pol 
yLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])] - I*PolyLog[2, Sqrt[1 + I*a*x] 
/Sqrt[1 - I*a*x]]))/(c*Sqrt[c + a^2*c*x^2]))/c
 

3.3.46.3.1 Defintions of rubi rules used

rule 208

Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), 
x] /; FreeQ[{a, b}, x]
 

rule 209

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
 

rule 5465

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ 
.), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 
1))), x] - Simp[b*(p/(2*c*(q + 1)))   Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) 
^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 
 0] && NeQ[q, -1]
 

rule 5489

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_ 
Symbol] :> Simp[(-2/Sqrt[d])*(a + b*ArcTan[c*x])*ArcTanh[Sqrt[1 + I*c*x]/Sq 
rt[1 - I*c*x]], x] + (Simp[I*(b/Sqrt[d])*PolyLog[2, -Sqrt[1 + I*c*x]/Sqrt[1 
 - I*c*x]], x] - Simp[I*(b/Sqrt[d])*PolyLog[2, Sqrt[1 + I*c*x]/Sqrt[1 - I*c 
*x]], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
 

rule 5493

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2 
]), x_Symbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]   Int[(a + b*ArcTan 
[c*x])^p/(x*Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[ 
e, c^2*d] && IGtQ[p, 0] &&  !GtQ[d, 0]
 

rule 5501

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 
)^(q_), x_Symbol] :> Simp[1/d   Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c 
*x])^p, x], x] - Simp[e/d   Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*x]) 
^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2* 
q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
 

3.3.46.4 Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.33

method	result	size
default	\(-\frac {\left (i+3 \arctan \left (a x \right )\right ) \left (i a^{3} x^{3}+3 a^{2} x^{2}-3 i a x -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{72 \left (a^{2} x^{2}+1\right )^{2} c^{3}}+\frac {5 \left (\arctan \left (a x \right )+i\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 c^{3} \left (a^{2} x^{2}+1\right )}-\frac {5 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )-i\right )}{8 c^{3} \left (a^{2} x^{2}+1\right )}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a^{3} x^{3}-3 a^{2} x^{2}-3 i a x +1\right ) \left (-i+3 \arctan \left (a x \right )\right )}{72 c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}+\frac {i \left (i \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-i \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c^{3}}\)	\(370\)

input

int(arctan(a*x)/x/(a^2*c*x^2+c)^(5/2),x,method=_RETURNVERBOSE)
 

output

-1/72*(I+3*arctan(a*x))*(I*a^3*x^3+3*a^2*x^2-3*I*a*x-1)*(c*(a*x-I)*(I+a*x) 
)^(1/2)/(a^2*x^2+1)^2/c^3+5/8*(arctan(a*x)+I)*(1+I*a*x)*(c*(a*x-I)*(I+a*x) 
)^(1/2)/c^3/(a^2*x^2+1)-5/8*(c*(a*x-I)*(I+a*x))^(1/2)*(I*a*x-1)*(arctan(a* 
x)-I)/c^3/(a^2*x^2+1)+1/72*(c*(a*x-I)*(I+a*x))^(1/2)*(I*a^3*x^3-3*a^2*x^2- 
3*I*a*x+1)*(-I+3*arctan(a*x))/c^3/(a^4*x^4+2*a^2*x^2+1)+I*(I*arctan(a*x)*l 
n((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-I*arctan(a*x)*ln(1-(1+I*a*x)/(a^2*x^2+1)^ 
(1/2))+polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-polylog(2,(1+I*a*x)/(a^2*x^ 
2+1)^(1/2)))*(c*(a*x-I)*(I+a*x))^(1/2)/(a^2*x^2+1)^(1/2)/c^3
 

3.3.46.5 Fricas [F]

\[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x} \,d x } \]

input

integrate(arctan(a*x)/x/(a^2*c*x^2+c)^(5/2),x, algorithm="fricas")
 

output

integral(sqrt(a^2*c*x^2 + c)*arctan(a*x)/(a^6*c^3*x^7 + 3*a^4*c^3*x^5 + 3* 
a^2*c^3*x^3 + c^3*x), x)
 

3.3.46.6 Sympy [F(-2)]

Exception generated. \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \]

input

integrate(atan(a*x)/x/(a**2*c*x**2+c)**(5/2),x)
 

output

Exception raised: TypeError >> Invalid comparison of non-real zoo
 

3.3.46.7 Maxima [F]

\[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x} \,d x } \]

input

integrate(arctan(a*x)/x/(a^2*c*x^2+c)^(5/2),x, algorithm="maxima")
 

output

integrate(arctan(a*x)/((a^2*c*x^2 + c)^(5/2)*x), x)
 

3.3.46.8 Giac [F]

\[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x} \,d x } \]

input

integrate(arctan(a*x)/x/(a^2*c*x^2+c)^(5/2),x, algorithm="giac")
 

output

sage0*x
 

3.3.46.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\mathrm {atan}\left (a\,x\right )}{x\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

input

int(atan(a*x)/(x*(c + a^2*c*x^2)^(5/2)),x)
 

output

int(atan(a*x)/(x*(c + a^2*c*x^2)^(5/2)), x)